Robustness of solutions of an infinite-dimensional algebraic Sylvester equation under bounded perturbations

• Autorzy: Emirsajłow Z.
• Języki publikacji: EN

Abstrakty

EN

The output regulation problem for distributed parameter systems, see e.g. Byrnes, Lauko, Gillian and Shubov (2000) and Paunonen (2011), and the observer design problem for such systems, see e.g. Emirsajłow (2012), are examples of import ant control problems, where analysis of infinite-dimensional algebraic Sylvester equations plays a crucial role. This paper studies bounded perturbations of the unbounded operators of the algebraic infinitedimensional Sylvester equation. We derive some estimate on the perturbation operator under which the algebraic Sylvester equation preserves a unique solution or, in the control systems terminology, a solution is robust under small bounded perturbations. In our approach we employ the concept of an implemented semigroup, see e.g. Alber (2001) and Emirsajłow (2012), which is a special case of the so-called bi-continuous semigroup, e.g. Farkas (2004).

Słowa kluczowe

EN

distributed parameter systems; algebraic Sylvester equation; bounded perturbations; implemented semigroup

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2013
• Tom: Vol. 42, no. 1
• Strony: 27--45
• Opis fizyczny: Bibliogr. 7 poz.

Twórcy

autor

Emirsajłow Z.

• Department of Control and Measurements West Pomeranian University of Technology in Szczecin Gen. Sikorskiego 37, 70-313 Szczecin, Poland

Bibliografia

• 1. Alber, J. (2001) On implemented semigroups. Semigroup Forum 63, 371–386.
• 2. Byrnes, C. I., Lauko, I. G., Gillian, D. S. and Shubov, V. I. (2000) Output regulation problem for linear distributed parameter systems. IEEE Transactions on Automatic Control 45, 2236–2252.
• 3. Emirsajłow, Z. (2012) Infinite-dimensional Sylvester equations: basic theory and applications to observer design. International Journal of Applied Mathematics and Computer Scienes 22, 245–257.
• 4. Emirsajłow, Z. and Townley, S. (2005) On application of the implemented semigroup to a problem arising in optimal control. International Journal of Control 78, 298–310.
• 5. Farkas, B. (2004) Perturbations of bi-continuous semigroups. Studia Mathematica 161, 147–161.
• 6. Paunonen, L. (2011) Output Regulation Theory for Linear Systems with Infinite- Dimensional and Periodic Exosystems. Tampere University of Technology, Tampere.
• 7. Phong, V. Q. (1991) The operator equation ax-xb=c with unbounded operators a and b and related Cauchy problems. Mathematische Zeitschrift 208, 567–588.